156
6 Interactions
for each half, a more detailed pattern can be discerned [25]. For the anti-bonding
orbitals, the general theorem, Eq. (6.132), can be applied directly. The result is illustrated in Fig. 6.9. By this theorem, the 3n/2 edge anti-bonds are split into two
subsets containing (1 + n/2) and (n − 1) orbitals. The former, higher lying, subset transforms as Γ − Γ ǫ . These terms correspond to circulations around the faces,
which means that these levels will be highly anti-bonding. In fact, they are always at
the top of the skeletal spectrum. Note that the pseudo-scalar term, Γ ǫ , does not take
part. This is because a uniform circulation around all faces in the same sense has no
contribution on the edges. Below this is a subset of weakly anti-bonding orbitals,
transforming as Γ σ (v) − Γ 0 . These orbitals are more localized on the vertices. The
Γ 0 term is not included since this is the totally-symmetric molecular orbital which
is completely bonding, and thus will appear in the lower half of the diagram.
Furthermore, the edge-bonding half can be analysed with the help of Eq. (6.139).
The 3n/2 edge bonds split into two subsets of dimension (n − 2) and (2 + n/2).
This analysis involves the fibre representation Γ σ (f ) × Γ T , which can be decomposed into a radial σ - and tangential π -part. The σ -part corresponds to cylindricallysymmetric bonds around the faces, and will thus be strongly bonding. For the π -part
the face terms contain a nodal plane through the faces, and thus will be less bonding.
Frontier Orbitals in Leapfrog Fullerenes
Fullerenes are trivalent polyhedra of carbon, consisting of hexagons and pentagons.
The following relations hold:
v − e + f = 2
3v = 2e
f 5 + f 6 = f
5f 5 + 6f 6 = 3v
(6.142)
The first two relationships are from Eqs. (6.128) and (6.140). The third expresses
that the total number of the faces is the sum of the number of pentagons (f 5 ), and
hexagons (f 6 ). The final equation indicates that by counting the hexagons six times,
and the pentagons five times, we have counted all vertices three times, since every vertex is at the junction of three faces. Even though there are fewer equations,
here, than unknowns, it can easily be seen by manipulation of Eq. (6.142) that the
only value that the number of pentagons, f 5 , can take on is 12. Hence, the smallest
fullerene is the dodecahedron C 20 , which only consists of pentagons. Also note that
the number of atoms in a fullerene must be even, since 3v must be divisible by 2, as
e is an integer. Taking the leapfrog, L, of a primitive fullerene, P , is an operation of
cage expansion, which yields a fullerene with three times as many atoms [26]. This
procedure is described by the following rule:
L = Dual(OmnicapP)
(6.143)
6 Interactions
for each half, a more detailed pattern can be discerned [25]. For the anti-bonding
orbitals, the general theorem, Eq. (6.132), can be applied directly. The result is illustrated in Fig. 6.9. By this theorem, the 3n/2 edge anti-bonds are split into two
subsets containing (1 + n/2) and (n − 1) orbitals. The former, higher lying, subset transforms as Γ − Γ ǫ . These terms correspond to circulations around the faces,
which means that these levels will be highly anti-bonding. In fact, they are always at
the top of the skeletal spectrum. Note that the pseudo-scalar term, Γ ǫ , does not take
part. This is because a uniform circulation around all faces in the same sense has no
contribution on the edges. Below this is a subset of weakly anti-bonding orbitals,
transforming as Γ σ (v) − Γ 0 . These orbitals are more localized on the vertices. The
Γ 0 term is not included since this is the totally-symmetric molecular orbital which
is completely bonding, and thus will appear in the lower half of the diagram.
Furthermore, the edge-bonding half can be analysed with the help of Eq. (6.139).
The 3n/2 edge bonds split into two subsets of dimension (n − 2) and (2 + n/2).
This analysis involves the fibre representation Γ σ (f ) × Γ T , which can be decomposed into a radial σ - and tangential π -part. The σ -part corresponds to cylindricallysymmetric bonds around the faces, and will thus be strongly bonding. For the π -part
the face terms contain a nodal plane through the faces, and thus will be less bonding.
Frontier Orbitals in Leapfrog Fullerenes
Fullerenes are trivalent polyhedra of carbon, consisting of hexagons and pentagons.
The following relations hold:
v − e + f = 2
3v = 2e
f 5 + f 6 = f
5f 5 + 6f 6 = 3v
(6.142)
The first two relationships are from Eqs. (6.128) and (6.140). The third expresses
that the total number of the faces is the sum of the number of pentagons (f 5 ), and
hexagons (f 6 ). The final equation indicates that by counting the hexagons six times,
and the pentagons five times, we have counted all vertices three times, since every vertex is at the junction of three faces. Even though there are fewer equations,
here, than unknowns, it can easily be seen by manipulation of Eq. (6.142) that the
only value that the number of pentagons, f 5 , can take on is 12. Hence, the smallest
fullerene is the dodecahedron C 20 , which only consists of pentagons. Also note that
the number of atoms in a fullerene must be even, since 3v must be divisible by 2, as
e is an integer. Taking the leapfrog, L, of a primitive fullerene, P , is an operation of
cage expansion, which yields a fullerene with three times as many atoms [26]. This
procedure is described by the following rule:
L = Dual(OmnicapP)
(6.143)